Refinements and higher-order beliefs: a unified survey

Abstract

This paper presents a simple framework that allows us to survey and relate some different strands of the game theory literature. We describe a “canonical” way of adding incomplete information to a complete information game. This framework allows us to give a simple “complete theory” interpretation (Kreps in Game theory and economic modelling. Clarendon Press, Oxford, 1990) of standard normal form refinements such as perfection, and to relate refinements both to the “higher-order beliefs literature” (Rubinstein in Am Econ Rev 79:385–391, 1989; Monderer and Samet in Games Econ Behav 1:170–190, 1989; Morris et al. in Econ J Econ Soc 63:145–157, 1995; Kajii and Morris in Econ J Econ Soc 65:1283–1309, 1997a) and the “payoff uncertainty approach” (Fudenberg et al. in J Econ Theory 44:354–380, 1988; Dekel and Fudenberg in J Econ Theory 52:243–267, 1990).

On the generic robustness of solution concepts to incomplete information

“…In contrast to the above robust lack of robustness of Nash equilibrium to incomplete information, Weinstein and Yildiz (2007) have shown that interim correlated rationalizable actions are generically robust to players' beliefs. The relevance of this result to those of Kajii and Morris (1997b) is that, while the entire hierarchy of players' beliefs is not explicitly part of the notion of robustness used by Kajii and Morris (1997b), players' beliefs are central to their analysis and results. Thus, Weinstein and Yildiz's (2007) paper is a natural continuation of Kajii and Morris's (1997b) work of analyzing the robustness of equilibria to incomplete information.…”

“…Amongst those robustness problems, the one concerning the robustness of Nash equilibrium of complete information games to incomplete information has received considerable attention. Specifically, Kajii and Morris (1997b), and Carlsson and van Damme (1993) before, have shown that some Nash equilibria of some normal-form games are not robust to incomplete information. This lack of robustness of Nash equilibrium of complete information games to incomplete information is particularly striking because it holds for some strict Nash equilibria, which were thought to be robust to all sorts of perturbations.…”

“…This result is used to show why interim correlated rationalizable actions are generically robust to players' hierarchies of beliefs as established by Weinstein and Yildiz (2007) even without their richness condition. It is also used to provide a formal sense according to which, in the setting of Kajii and Morris (1997b), the set of representations of a complete information game is small in the space of incomplete information games in which it is embedded. This difference in relative sizes makes the robustness problems of Kajii and Morris (1997b) and Weinstein and Yildiz (2007) be incomparable and helps explaining why their conclusions differ so significantly.…”

“…It is also used to provide a formal sense according to which, in the setting of Kajii and Morris (1997b), the set of representations of a complete information game is small in the space of incomplete information games in which it is embedded. This difference in relative sizes makes the robustness problems of Kajii and Morris (1997b) and Weinstein and Yildiz (2007) be incomparable and helps explaining why their conclusions differ so significantly. * I wish to thank Christian Hellwig, Atsushi Kajii, Mário Páscoa, Satoru Takahashi and two anonymous referees for very helpful comments.…”

A note on identification of discrete choice models for bundles and binary games

“…These results are remarkable because this class of games often has a large number of equilibria. Further, Ui (2001) shows that if the potential maximizer is unique, then this equilibrium is robust in the sense of Kajii and Morris (1997).…”